| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 6·11-s + 14-s − 16-s − 2·19-s − 6·22-s − 28-s − 6·29-s + 2·31-s + 5·32-s + 8·37-s − 2·38-s − 2·41-s + 8·43-s + 6·44-s + 4·47-s + 49-s + 6·53-s − 3·56-s − 6·58-s + 6·61-s + 2·62-s + 7·64-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 1.80·11-s + 0.267·14-s − 1/4·16-s − 0.458·19-s − 1.27·22-s − 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.883·32-s + 1.31·37-s − 0.324·38-s − 0.312·41-s + 1.21·43-s + 0.904·44-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.400·56-s − 0.787·58-s + 0.768·61-s + 0.254·62-s + 7/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.822405805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.822405805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.95694516860674, −12.36523891460884, −12.11384095421735, −11.43705539541126, −10.92576206107609, −10.61387933465316, −10.03584349373437, −9.602944174379397, −9.004621031742368, −8.655757959549391, −8.035064521079281, −7.662278340805987, −7.325142084400685, −6.402249522667036, −6.036636615516293, −5.501141728749685, −5.091522725793627, −4.725734713175124, −4.076587871875765, −3.712793721978571, −2.951269803435960, −2.477981471223433, −2.073290895304537, −0.9832254729085333, −0.3716438696648349,
0.3716438696648349, 0.9832254729085333, 2.073290895304537, 2.477981471223433, 2.951269803435960, 3.712793721978571, 4.076587871875765, 4.725734713175124, 5.091522725793627, 5.501141728749685, 6.036636615516293, 6.402249522667036, 7.325142084400685, 7.662278340805987, 8.035064521079281, 8.655757959549391, 9.004621031742368, 9.602944174379397, 10.03584349373437, 10.61387933465316, 10.92576206107609, 11.43705539541126, 12.11384095421735, 12.36523891460884, 12.95694516860674
